A sequence with a second difference is a quadratic sequence. This means that the difference between consecutive terms in the sequence is not constant, but the difference between those differences is constant.
Understanding First and Second Differences
To understand a second difference, let's break down the concepts:
- First Difference: The difference between each term and the one that precedes it. For example, in the sequence 1, 4, 9, 16..., the first differences are 3, 5, and 7.
- Second Difference: The difference between the first differences. Using the same example, the differences between 3, 5, and 7 are 2 and 2. This constant value is the second difference.
Characteristics of a Quadratic Sequence
Here are the key characteristics of a quadratic sequence:
- Non-constant First Differences: The differences between the terms in the sequence are not consistent.
- Constant Second Differences: The differences between the first differences are consistent, creating a constant second difference.
According to the provided reference, any sequence that has a common second difference is a quadratic sequence. Also, the first differences of a quadratic sequence form a sequence and this sequence has a constant difference between consecutive terms.
Examples
Let's see how to identify a quadratic sequence with a common second difference:
| Sequence | First Difference | Second Difference |
|---|---|---|
| 1, 4, 9, 16, 25... | 3, 5, 7, 9... | 2, 2, 2... |
| 2, 8, 18, 32, 50... | 6, 10, 14, 18... | 4, 4, 4... |
As you can see, in each example above, the second difference is constant, indicating they are indeed quadratic sequences.
Practical Insights
- If you come across a sequence where the first differences are not constant but do form a sequence with a common difference, you are dealing with a quadratic sequence.
- Knowing that a sequence has a constant second difference allows you to predict further terms in that sequence.
Conclusion
A sequence with a second difference is a quadratic sequence, which is characterized by a non-constant first difference and a constant second difference. The first differences themselves form a sequence with a constant difference.
